Having been to a NATS TRUCE day the answer in the UK is most definitely wait for a clearance.
One of their closest "Near Misses" of last year was caused by just the situation you are describing. A very busy London TMA, lots of wx avoidance and lots of stepping on each other. A Biz Jet decided to avoid without getting a clearance and they caused a serious loss of separation with an Air Lingus on departure from Heathrow. (And of course someone saying TCAS RA is yet more talking and confusion on the frequency)!
The word from NATS is that if you can't get a word in, and you MUST turn to avoid WX, you should squawk 7700. This will cause you to show red on their screens. They will stop all other communication and ask you what the problem is. You can then tell them you are turning due to weather.
Your target going red will also immediately draw their attention to you and enable them to move aircraft out of your way. Controllers operating the sectors above or below the sector that you are in will automatically stop any aircraft climbing or descending through your level when they see a 7700.
I admit it does seem strange squawking 7700 when you are not on a Mayday, but this is what NATS want us to do.
I'd also highly recommend going to a TRUCE day. It's a real eye opener and very interesting.
SW